Moduli of Smoothness and Rate of a.e. Convergence for Some Convolution Operators

Abstract

One purpose of this chapter is to establish results on the rate of almost everywhere convergence of approximation processes of convolution type in Lp(ℝn), where instead of a particular rate (like tμ, μ > 0, t → 0+), fractional moduli of smoothness are employed. An essential tool is a modified K-functional. Away from saturation orders these results are nearly optimal. A second purpose is to illustrate that the methods applied also work in other settings which feature a convolution/multiplier structure. © Springer Science+Business Media, LLC 2013.